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Properties

Label 2-72-72.11-c3-0-9
Degree $2$
Conductor $72$
Sign $-0.367 - 0.929i$
Analytic cond. $4.24813$
Root an. cond. $2.06110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.44 + 1.41i)2-s + (−1.27 + 5.03i)3-s + (3.99 + 6.92i)4-s + (−10.2 + 10.5i)6-s + 22.6i·8-s + (−23.7 − 12.8i)9-s + (17.1 + 9.88i)11-s + (−39.9 + 11.3i)12-s + (−32.0 + 55.4i)16-s − 24.2i·17-s + (−39.9 − 65.0i)18-s + 163.·19-s + (27.9 + 48.4i)22-s + (−113. − 28.8i)24-s + (62.5 − 108. i)25-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (−0.245 + 0.969i)3-s + (0.499 + 0.866i)4-s + (−0.697 + 0.716i)6-s + 0.999i·8-s + (−0.879 − 0.475i)9-s + (0.469 + 0.270i)11-s + (−0.962 + 0.272i)12-s + (−0.500 + 0.866i)16-s − 0.345i·17-s + (−0.523 − 0.851i)18-s + 1.97·19-s + (0.270 + 0.469i)22-s + (−0.969 − 0.245i)24-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $-0.367 - 0.929i$
Analytic conductor: \(4.24813\)
Root analytic conductor: \(2.06110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 72,\ (\ :3/2),\ -0.367 - 0.929i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.20299 + 1.76969i\)
\(L(\frac12)\) \(\approx\) \(1.20299 + 1.76969i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.44 - 1.41i)T \)
3 \( 1 + (1.27 - 5.03i)T \)
good5 \( 1 + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-17.1 - 9.88i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 24.2iT - 4.91e3T^{2} \)
19 \( 1 - 163.T + 6.85e3T^{2} \)
23 \( 1 + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + (367. - 211. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-281. + 488. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + (775. - 447. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-456. - 790. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 799.T + 3.89e5T^{2} \)
79 \( 1 + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (590. + 340. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.32e3iT - 7.04e5T^{2} \)
97 \( 1 + (-455. + 788. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.51359244432649909510511916003, −13.71978473051851101024449144556, −12.16995897536698173500056896225, −11.44682160465242634468696457744, −10.03213296201352302791187877818, −8.718650027025751772157131368631, −7.15263739756811807273544871899, −5.71766680353695040591453081189, −4.59594841860295216633188778804, −3.20792584697725461923329865018, 1.31300777856353274923564530339, 3.18536451190689298798195615372, 5.18382818282928435609585116713, 6.36785492599646676855443833224, 7.58532455796973039851646219968, 9.378876588165123965930891414532, 10.94534871210637133255857097787, 11.78065222386053409882174940688, 12.69224302061168929919210283819, 13.70280996415659990644579777657

Graph of the $Z$-function along the critical line